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Theorem conncompss 22027
Description: The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
conncomp.2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
Assertion
Ref Expression
conncompss ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑆)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋
Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem conncompss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simp1 1133 . . . . 5 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑋)
2 conntop 22011 . . . . . . 7 ((𝐽t 𝑇) ∈ Conn → (𝐽t 𝑇) ∈ Top)
323ad2ant3 1132 . . . . . 6 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝐽t 𝑇) ∈ Top)
4 restrcl 21751 . . . . . . 7 ((𝐽t 𝑇) ∈ Top → (𝐽 ∈ V ∧ 𝑇 ∈ V))
54simprd 499 . . . . . 6 ((𝐽t 𝑇) ∈ Top → 𝑇 ∈ V)
6 elpwg 4523 . . . . . 6 (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
73, 5, 63syl 18 . . . . 5 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
81, 7mpbird 260 . . . 4 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 ∈ 𝒫 𝑋)
9 3simpc 1147 . . . 4 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn))
10 eleq2 2904 . . . . . 6 (𝑦 = 𝑇 → (𝐴𝑦𝐴𝑇))
11 oveq2 7146 . . . . . . 7 (𝑦 = 𝑇 → (𝐽t 𝑦) = (𝐽t 𝑇))
1211eleq1d 2900 . . . . . 6 (𝑦 = 𝑇 → ((𝐽t 𝑦) ∈ Conn ↔ (𝐽t 𝑇) ∈ Conn))
1310, 12anbi12d 633 . . . . 5 (𝑦 = 𝑇 → ((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) ↔ (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn)))
14 eleq2 2904 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
15 oveq2 7146 . . . . . . . 8 (𝑥 = 𝑦 → (𝐽t 𝑥) = (𝐽t 𝑦))
1615eleq1d 2900 . . . . . . 7 (𝑥 = 𝑦 → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t 𝑦) ∈ Conn))
1714, 16anbi12d 633 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
1817cbvrabv 3476 . . . . 5 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} = {𝑦 ∈ 𝒫 𝑋 ∣ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)}
1913, 18elrab2 3668 . . . 4 (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ↔ (𝑇 ∈ 𝒫 𝑋 ∧ (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn)))
208, 9, 19sylanbrc 586 . . 3 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
21 elssuni 4849 . . 3 (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} → 𝑇 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
2220, 21syl 17 . 2 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
23 conncomp.2 . 2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
2422, 23sseqtrrdi 4002 1 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  {crab 3136  Vcvv 3479  wss 3918  𝒫 cpw 4520   cuni 4819  (class class class)co 7138  t crest 16683  Topctop 21487  Conncconn 22005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673  df-rest 16685  df-top 21488  df-conn 22006
This theorem is referenced by:  conncompcld  22028  tgpconncompeqg  22706  tgpconncomp  22707
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